Problem: The moon's distance from Earth varies in a periodic way that can be modeled by a trigonometric function. When the moon is at its perigee (closest point to Earth), it's about $363{,}000 \text{ km}$ away. When it's at its apogee (farthest point from Earth), it's about $406{,}000\text{ km}$ away. The moon's apogees occur $27.3$ days apart. The moon will reach its apogee on January $2$, $2016$. Find the formula of the trigonometric function that models the distance $D$ between Earth and the moon $t$ days after January $1$, $2016$. Define the function using radians. $ D(t) = $
Solution: Let's start by writing a formula for the distance from the moon to the earth $u$ days after its apogee. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. The distance from the moon to the earth is at a maximum at time $u = 0$, so we'll use a cosine function to model the distance as a function of time. The distance from the moon to the earth has period $27.3$ days. The midline of the distance is halfway between its maximum value and its minimum value, or $\dfrac{406{,}000 + 363{,}000}{2} =384{,}500$ Its amplitude is half the difference between its maximum and minimum values, or $\dfrac{406{,}000 - 363{,}000}{2} = 21{,}500$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we can stretch it horizontally by a factor of ${\dfrac{27.3}{2\pi}}$, stretch it vertically by a factor of ${21{,}500}$, and move it up ${384{,}500}$ units: $ D(u) = {21{,}500}\cos\left({\dfrac{2\pi}{27.3}}u\right) + {384{,}500}$ Since the moon reaches its apogee on January $2$ which is $1$ day after January $1$, the day that is $t$ days after January $1$ is $t - 1$ days after the moon's apogee, so $u = t-1$ : $ D(t) = {21{,}500}\cos\left({\dfrac{2\pi}{27.3}}(t-1)\right) + {384{,}500}$ The function $ D(t) = {21{,}500}\cos\left({\dfrac{2\pi}{27.3}}(t-1)\right) + {384{,}500}$ has period $27.3$, amplitude $21{,}500$ and midline $y = 384{,}500$, and it reaches its maximum at time $t = 1$, so it's a good model of the distance from the earth to the moon.